We provide exact computations for the average stability term via the Sherman-Morrison formula (see e.g., [1]}), which states that for any \(j = 1, \ldots , n + 1\)  we have
\(\left(\tilde{\Sigma }_{\lambda }^{(-j)}\right)^{-1} =\tilde{\Sigma }_{\lambda }^{-1}+ \frac{\tilde{\Sigma }_{\lambda }^{-1}X_{j}X_{j}^{\mathsf {T}}\tilde{\Sigma }_{\lambda }^{-1}}{1 - X_j^{\mathsf {T}}\tilde{\Sigma }_{\lambda }^{-1}X_j}.\) 
